Introduction
I have a project on GitHub about Quantum Scattering1 on a Lennard-Jones potential. The idea is from chapter 2 of a book2 I already mentioned on this blog. For this project I won’t put theory here, but refer you to the book instead. It’s a very good book, it’s worth having. I don’t know a better computational physics book that is also treating many topics, the ones that are better are on narrow subjects, not so general. I strongly recommend it. I won’t present the theory here except a few words, I’ll let you look into the book and in a linked article. For the rest, there is the code…
Program in action
Here is a video on youtube:
The peaks you see there are due of resonant scattering states (orbiting resonances).
Theory description
What you see in the chart in the above video or in the featured picture is the total cross-section for scattering of a hydrogen atom on some noble gas atom. The scattering potential is approximated with a parametrized Lennard-Jones potential and taking advantage of the spherical symmetry, the Schrödinger equation, more specifically, the radial one, is solved using the Numerov method. Then the cross-section is computed using the phase shift. For the details you either will have to get the book, or look into one of those links: Central Potential Scattering and Phase Shifts3 or Scattering Tutorial4.
Now, the last but not the least, the paper that has some results reproduced by the program: Molecular beam scattering studies of orbiting resonances and the determination of van der Waals potentials for H–Ne, Ar, Kr, and Xe and for H2–Ar, Kr, and Xe5.
The code
The code is the best documentation. If interested only in the computational physics part, you have to look only in one namespace: Scattering, also perhaps in the SpecialFunctions namespace, but the last version uses the std implementations. The class names are very suggestive, LennardJonesPotential, Numerov and Scattering, to name some. I hope the code expresses intent well enough to not need much description.
Libraries used
As usual for the last projects, here they are: wxWidgets6 and VTK7 (no, not Eigen this time).
Conclusion
Hopefully this is not the last project for this blog that takes advantage of the spherical symmetry, but I won’t reveal more of my plans about it.
As usual, if you find bugs or have suggestions, please comment.
- Scattering project on GitHub ↩
- Computational Physics book by Jos Thijssen ↩
- Central Potential Scattering and Phase Shifts Some page from Florida State University ↩
- Scattering Tutorial by Erich Mueller ↩
- Molecular beam scattering studies of orbiting resonances and the determination of van der Waals potentials for H–Ne, Ar, Kr, and Xe and for H2–Ar, Kr, and Xe published in 1979 in J. Chem. Phys., authors J. Peter Toennies, Wolfgang Welz, and Günther Wolf ↩
- wxWidgets Cross-platform GUI library ↩
- VTK, The Visualization Toolkit The scientific (and not only) visualization library ↩
